Plus Blog

September 24, 2006
Monday, September 25, 2006

Celebrating mathematics at work

The relevance and usefulness of mathematics is most clearly demonstrated in its application to real-world problems, many of which have featured in Plus over the years. To celebrate mathematics at work in the real-world, the International Council for Industrial and Applied Mathematics have announced the winners of the five ICIAM prizes for 2007.

The Pioneer Prize, recognising innovations in applied mathematics, was jointly won by Ingrid Daubechies (Princeton University, USA) and Heinz Engl (Johannes Kepler Universität Linz, Austria). Daubechies' work on wavelets has found widespread use in image processing and frequency analysis. Engl was recognised for his work on inverse problems to the solution of a wide range of industrial problems and for his promotion world-wide of industrial and applied mathematics.

The Collatz Prize is awarded to outstanding applied mathematicians under the age of 42. It was won by Felix Otto (Universität Bonn, Germany) for his fundamental contributions in areas ranging from micromagnetics to mass transportation problems.

Joseph Keller (Stanford University, USA) won the Lagrange Prize for his exceptional contributions to the field. Not only has he influenced the course of modern applied mathematics over the last 50 years, he has also had a great impact in pure mathematics as well. He developed the Geometrical Theory of Diffractions that provided the first systematic description of wave propagation around edges and corners of an obstacle. It has been widely used for radar reflection from targets, elastic wave scattering from defects in solids, acoustic wave propagation in the ocean radar and many other fields. He has also made contributions to quantum mechanics, optics, acoustics, biophysics and biomechanics and transport theory.

Peter Deuflhard (ZIB Berlin, Germany), winner of the Maxwell Prize, is one of the founders of modern scientific computing, and is described as having made a contribution to applied mathematics that is without parallel, including applications in chemical engineering, medicine and biotechnology, microwave technology and nano-optics.

The Su Buchin Prize is more unusual in the field of mathematics, recognising the "application of Mathematics to emerging economies and human development". The winner, Gilbert Strang (MIT, USA), is one of the most recognised mathematicians in the developing world through his promotion of mathematical research and education across the globe. As well as numerous visits himself, he has organised visits by other mathematicians to developing countries, and his educational materials are available on the web, free-of-charge to any user anywhere in the world, through MIT's OpenCourseWare. Alongside his contributions in bringing mathematics to people around the world, his research has made contributions in many areas of pure and applied mathematics.

The prizes will be presented at the International Congress for Industrial and Applied Mathematics in Zürich next year, a major international celebration of mathematics in action. More information on the prizes, and the winners, can be found at the ICIAM website. And you can read more about industrial mathematics on Plus.

posted by Plus @ 5:04 PM

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September 18, 2006
Tuesday, September 19, 2006

Einstein passes with 99.5% result

On September 14th, after 3 years observing a rare double pulsar, an international team led by Prof. Michael Kramer from the Jodrell Bank Observatory announced staggering results confirming Einstein's general theory of relativity to 99.5% accuracy. The observations were made on a pair of pulsars orbiting each other approximately 2000 light years from Earth. Each pulsar weighs a little more than the Sun but has a diameter of around 10km. The extreme mass of the pair gives us a rare opportunity to study the predicted effects of general relativity.

Sometimes when a large star collapses it becomes a very small, very dense object known as a neutron star. A pulsar is a rotating neutron star. Pulsars are extremely useful for astronomical observations as they emit constant, powerful beams of radio waves similar to how a lighthouse emits a beam of light. Each time a beam sweeps past Earth we observe a distinctive radio pulse. By precisely measuring the time between pulses we can make detailed calculations on the effect gravity is having on the radio waves. By current theory, this behaviour should be governed by the laws of general relativity and it turns out that the predictions match with observation to an astonishing level of accuracy.

Another very important result is that the distance between the two pulsars is shrinking by around 7mm per day. This also agrees with Einstein's predictions. According to general relativity, the pulsars should be emitting gravitational waves and although these waves have never been directly detected, their emission should cause the pulsar system to lose energy. The observable effect of this would be the pulsars spiraling towards each other by precisely the amount observed giving compelling evidence for the existence of gravitational waves.

However Dr. Kramer believes that there are still many more exciting results to come: "The double pulsar is really quite an amazing system. It not only tells us a lot about general relativity, but it is a superb probe of the extreme physics of super-dense matter and strong magnetic fields but is also helping us to understand the complex mechanisms that generate the pulsar's radio beacons." He concludes; "We have only just begun to exploit its potential!"

You can read the full story at the Jodrell Bank Observatory website.

posted by Plus @ 2:06 PM

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September 4, 2006
Tuesday, September 05, 2006

School maths to beat climate change

Good maths and science school education is vital in dealing with climate change according to Frances Cairncross, President of the British Asoociation of the Advancement of Science and Chair of Britain's Economic and Social Research Council. Speaking at the BA Festival of Science taking place in Norwich last week, Ms Cairncross said that trying to mitigate the impact of climate change is no longer sufficient: we need to put equal effort into preparing for a world with a different climate. "We need to think now about policies that prepare for a hotter, drier world, especially in poorer countries. That may involve, for instance, developing new crops, constructing flood defences, setting different building regulations, or banning building close to sea level."

To develop such strategies we don't only need scientists, but also a well-informed public that can partake in public debate and help to drive change. A good school education in maths and science is vital in creating both. "An innumerate population is less likely to devise good solutions to climate change and a host of other environmental problems than one at home with mathematical and scientific concepts," Ms Cairncross said. "Perhaps we need to experiment with a bounty for every A grade maths A-level taken in a maintained school... or a couple of extra UCAS points for each A grade."

Climate change features strongly at this year's Science Festival and the scientists' predictions are dire, giving us only a few years to seriously curb our carbon emissions or face drastic change. Ms Cairncross called the Kyoto agreement to cut greenhouse gas emissions "ineffectual" and suggested that an effective strategy to deal with environmental problems involves "persuading this generation to accept sacrifices on behalf of posterity." Such altruistic behaviour is only likely to come about if people understand the risks and realities involved in climate change and for this they need a sound mathematical and scientific grounding.

posted by Plus @ 12:52 PM

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August 28, 2006
Tuesday, August 29, 2006

Shining new light on dark matter

In the 1930s, astronomers discovered that many galaxy clusters observable from Earth have a much stronger gravitational field than they should have given their predicted mass. Further astronomical observations only added to this puzzle. After much consideration, it was concluded that something mysterious called dark matter must be involved. Dark matter is in all respects invisible and can only be detected by its gravitational effect on normal matter. If this new theory was right then dark matter would make up most of the mass of the universe.

However, in February three scientists claimed that dark matter was not necessary and in fact by slightly altering Einstein's equations for general relativity they could account for the acceleration. Not everyone was convinced by the new explanation though, and now new evidence has been put forward in support of dark matter through studying the "bullet" galaxy cluster with the Chandra X-ray telescope.

The cluster was created when two separate clusters smashed together. Tremendous amounts of energy were released in this collision; enough in fact to tear the normal matter and the dark matter apart. Even though dark matter is invisible, scientists were able to see the effect by measuring how the mass of the cluster was distributed.

The data gathered supported a model involving dark matter but not an altered form of general relativity as was previously proposed. No doubt the argument over the existence of dark matter will continue but supporters of the dark matter model believe this provides the most conclusive evidence yet.

You can read the full story on Science Daily

posted by Plus @ 2:37 PM

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August 16, 2006
Thursday, August 17, 2006

Conjecture to theorem to fame to fortune

The buzz is building in the mathematical community. It looks more and more likely that Grigori Perelman's proof of the Poincaré conjecture is correct — and that he has solved a problem that has eluded the best mathematical minds for more than a century. When Perelman first posted his proof on the web in 2002 many thought this would be just another failed attempt, but since then it has survived intense mathematical scrutiny and appears to close to being accepted as correct.

Now the rumour mill has gone into overdrive. Word on the mathematical street is that he will receive the Fields Medal (thought of as the maths equivalent to the Nobel prize) next week at the International Congress of Mathematics in Madrid. And not only mathematical glory awaits him. The Poincaré Conjecture is one of the seven Millennium Problems named by the Clay Institute, and if Perelman has proved it he is eligible for the $1 million prize. So if the rumours are right, Perelman's fame and fortune are just around the corner.

That is all very exciting, but there is something that may get Perelman even more column inches in the press (he has already made the front pages): Perelman has a history of not accepting prizes. It seems that not only may he refuse the $1 million Clay prize, he may refuse the Fields medal too. This would make him the first to refuse the Field's Medal, and the first not only to win the Clay prize, but also the first to turn it down.

Regardless of whether Perelman accepts the accolades that may come his way, the biggest news for most mathematicians is whether his work is finally accepted as correct — and whether we can start calling the Poincaré Conjecture, the Poincaré Theorem, after all this time.

The world of mathematics waits with baited breath....

posted by Plus @ 1:31 PM

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August 1, 2006
Wednesday, August 02, 2006

How to board a plane relatively fast

You'd think that boarding a plane is an easy thing to do: get on, find your seat and sit down. But reality is never like that: there's always that woman whose oversized make-up bag doesn't fit into the overhead locker, the business man who has to fold up his jacket with utmost precision and the family of five that try out every possible seating arrangement before settling down. But now some new research, reported in the New Scientist last week, shows that it's not all down to a few annoying individuals. "Enplaning", as airlines call it, really is a complicated business and it takes some complicated maths to model it: Einstein's theory of relativity.

Einstein's theory postulates that time passes differently depending on how you move through space: a person travelling in a space rocket at high speed, for example, will have aged less on his or her return than the people who stayed back on Earth (see Plus article What's so special about special relativity?). The key for airline boarding lies in the behaviour of an object in free-fall: Einstein's theory predicts that it will follow the path that takes longest to travel, where the time is measured from the point of view of the moving object.

Eitan Bachmat and his team from the Ben-Gurion University of Negev in Israel realised that, even though plane passengers usually aren't in free-fall, airline boarding involves maximisation of time in a way that can be modelled by Einstein's theory for free-falling objects. They applied their model not to the usual four-dimensional space (three space dimension and one time dimension) but to a new two-dimensional space based on the passengers' seat allocations and their position in the queue.

Having devised their model, the scientist checked to see if boarding passengers in a certain order, for example those in the last rows first, can make boarding any quicker. What they found is that the "last rows first" technique adopted by many airlines is no better than seating passengers randomly. In fact, getting passengers to queue in a random order is surprisingly efficient. The best option, in terms of boarding time, would be to assign to each passenger a specific place in the queue, but this is rather unrealistic as passengers are unlikely to respond well to such regimental techniques. In practise, the most efficient way of queuing takes into account the order within the rows: getting window seat passengers to board first and isle seat passengers to board last seems to work pretty well. The scientists also found that the time it takes to board a plane is proportional to the square root of the number of passengers.

This is the first application of Einstein's theory outside of physics and, according to Bachmat, one of the first scientific studies of airline boarding. So far, airlines have taken preciously little notice of the scientific studies, rather surprisingly given the great amount of money hinging on turn-around time of planes. But what many people would really like to know is whether the all-out-chaos approach of budget airlines that don't allocate seats is any more efficient than traditional ways of boarding. Unfortunately, Bachmat's model doesn't cover this. It's a whole different dimension.

posted by Plus @ 1:30 PM

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